Multiply Quantized Vortices in Fermionic Superfluids: Angular Momentum, Unpaired Fermions, and Spectral Asymmetry

We compute the orbital angular momentum $L_z$ of an $s$-wave paired superfluid in the presence of an axisymmetric multiply quantized vortex. For vortices with a winding number $|k|>1$, we find that in the weak-pairing BCS regime, $L_z$ is significantly reduced from its value $\hslash Nk/2$ in the Bose-Einstein condensation (BEC) regime, where $N$ is the total number of fermions. This deviation results from the presence of unpaired fermions in the BCS ground state, which arise as a consequence of spectral flow along the vortex subgap states. We support our results analytically and numerically by solving the Bogoliubov–de Gennes equations within the weak-pairing BCS regime.

Figure 1

Summary of main result: (a) for an elementary vortex ($k=1$), the fermionic spectrum has a vanishing spectral asymmetry and thus, all fermions are paired in the ground state, resulting in $L_z=\hslash N/2$ in the BCS regime. (b) In stark contrast, for an MQV ($k=2$ pictured here as an example), midgap states confined to the vortex core induce a nontrivial spectral asymmetry, which leads to unpaired fermions in the ground state. These reduce $L_z$ from its naïve value $\hslash N$ by an amount that scales quadratically with the splitting between the red branches.

Figure 2

BdG solution for MQVs with ${\mathrm{\Delta }}_0=0.15E_F$, $\mu =E_F$, and $k_{F}R=80$: (a) comparison of energy spectrum for $k=2$ with analytic approximation (in red); (b) spectral asymmetry for $k=2$; (c) energy spectrum for $k=3$ and (d) for $k=4$.

Figure 3

The analytic prediction $L_z/N=1-\alpha (\xi /R{)}^2$ (red line) fits the numerical data (blue dots) well over a wide window within the BCS regime $0.05\lesssim {\mathrm{\Delta }}_0/E_F\lesssim 0.25$ for an MQV with $k=2$. The slope of the fit equals two as shown on a log-log plot.